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Q7:
Integrate the functions in Exercises 1 to 23.
7. $\frac{x-1}{\sqrt{x^2-1}}$
Solution :
Step 1: Decomposition of the Integrand
The given integral is $I = \int \frac{x-1}{\sqrt{x^2-1}} dx$. We split the integral into two distinct parts to facilitate different integration techniques [By the linearity property of integration]:
$I = \int \frac{x}{\sqrt{x^2-1}} dx - \int \frac{1}{\sqrt{x^2-1}} dx$
Let $I_1 = \int \frac{x}{\sqrt{x^2-1}} dx$ and $I_2 = \int \frac{1}{\sqrt{x^2-1}} dx$. Thus, $I = I_1 - I_2$.
Step 2: Solving $I_1$ via Substitution
For $I_1 = \int \frac{x}{\sqrt{x^2-1}} dx$, we utilize the method of substitution [By the substitution rule for indefinite integrals]:
- Let $u = x^2 - 1$.
- Differentiating with respect to $x$, $du = 2x \, dx$, which implies $x \, dx = \frac{du}{2}$.
$I_1 = \int \frac{1}{\sqrt{u}} \cdot \frac{du}{2} = \frac{1}{2} \int u^{-1/2} du$
Applying the power rule $\int u^n du = \frac{u^{n+1}}{n+1} + C$:
$I_1 = \frac{1}{2} \left( \frac{u^{1/2}}{1/2} \right) = \sqrt{u} + C_1 = \sqrt{x^2-1} + C_1$
Step 3: Solving $I_2$ via Standard Formula
For $I_2 = \int \frac{1}{\sqrt{x^2-1}} dx$, we apply the standard integral formula [Per the table of integrals for algebraic forms]:
$\int \frac{1}{\sqrt{x^2-a^2}} dx = \ln|x + \sqrt{x^2-a^2}| + C$
Given $a = 1$, we have:
$I_2 = \ln|x + \sqrt{x^2-1}| + C_2$
Step 4: Synthesis of the Final Result
Combining the results from Step 2 and Step 3 into our expression from Step 1:
$I = I_1 - I_2 = \sqrt{x^2-1} - \ln|x + \sqrt{x^2-1}| + C$
Final Solution: $\sqrt{x^2-1} - \ln|x + \sqrt{x^2-1}| + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.4
- Q1: Integrate the functions in Exercises 1 to 23. 1. $\frac{3x^2}{x^6+1}$
- Q2: Integrate the functions in Exercises 1 to 23. 2. $\frac{1}{\sqrt{1+4x^2}}$
- Q3: Integrate the functions in Exercises 1 to 23. 3. $\frac{1}{\sqrt{(2-x)^2+1}}$
- Q4: Integrate the functions in Exercises 1 to 23. 4. $\frac{1}{\sqrt{9-25x^2}}$
- Q5: Integrate the functions in Exercises 1 to 23. 5. $\frac{3x}{1+2x^4}$
- Q6: Integrate the functions in Exercises 1 to 23. 6. $\frac{x^2}{1-x^6}$
- Q8: Integrate the functions in Exercises 1 to 23. 8. $\frac{x^2}{\sqrt{x^6+a^6}}$
- Q9: Integrate the functions in Exercises 1 to 23. 9. $\frac{\sec^2 x}{\sqrt{\tan^2 x + 4}}$
- Q10: Integrate the functions in Exercises 1 to 23. 10. $\frac{1}{\sqrt{x^2+2x+2}}$
- Q11: Integrate the functions in Exercises 1 to 23. 11. $\frac{1}{9x^2+6x+5}$
- Q12: Integrate the functions in Exercises 1 to 23. 12. $\frac{1}{\sqrt{7-6x-x^2}}$
- Q13: Integrate the functions in Exercises 1 to 23. 13. $\frac{1}{\sqrt{(x-1)(x-2)}}$
- Q14: Integrate the functions in Exercises 1 to 23. 14. $\frac{1}{\sqrt{8+3x-x^2}}$
- Q15: Integrate the functions in Exercises 1 to 23. 15. $\frac{1}{\sqrt{(x-a)(x-b)}}$
- Q16: Integrate the functions in Exercises 1 to 23. 16. $\frac{4x+1}{\sqrt{2x^2+x-3}}$
- Q17: Integrate the functions in Exercises 1 to 23. 17. $\frac{x+2}{\sqrt{x^2-1}}$
- Q18: Integrate the functions in Exercises 1 to 23. 18. $\frac{5x-2}{1+2x+3x^2}$
- Q19: Integrate the functions in Exercises 1 to 23. 19. $\frac{6x+7}{\sqrt{(x-5)(x-4)}}$
- Q20: Integrate the functions in Exercises 1 to 23. 20. $\frac{x+2}{\sqrt{4x-x^2}}$
- Q21: Integrate the functions in Exercises 1 to 23. 21. $\frac{x+2}{\sqrt{x^2+2x+3}}$
- Q22: Integrate the functions in Exercises 1 to 23. 22. $\frac{x+3}{x^2-2x-5}$
- Q23: Integrate the functions in Exercises 1 to 23. 23. $\frac{5x+3}{\sqrt{x^2+4x+10}}$
- Q24: Choose the correct answer in Exercises 24 and 25. 24. $\int \frac{dx}{x^2+2x+2}$ equals
- Q25: Choose the correct answer in Exercises 24 and 25. 25. $\int \frac{dx}{\sqrt{9x-4x^2}}$ equals
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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